---
book: 3
number: 18
id: "III.18"
kind: "theorem"
uses: ["[[book-1/proposition-17]]", "[[book-1/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.18"
license: "CC-BY-SA-4.0"
---

# III.18

*If a straight line touch a circle*, *and a straight line be joined from the centre to the point of contact*, *the straight line so joined will be perpendicular to the tangent*.

## Proof

For let a straight line *DE* touch the circle *ABC* at the point *C*, let the centre *F* of the circle *ABC* be taken, and let *FC* be joined from *F* to *C*; I say that *FC* is perpendicular to *DE*.

For, if not, let *FG* be drawn from *F* perpendicular to *DE*.

Then, since the angle *FGC* is right, the angle *FCG* is acute; [[book-1/proposition-17|I. 17]] and the greater angle is subtended by the greater side; [[book-1/proposition-19|I. 19]] therefore *FC* is greater than *FG*.

But *FC* is equal to *FB*; therefore *FB* is also greater than *FG*, the less than the greater: which is impossible.

Therefore *FG* is not perpendicular to *DE*.

Similarly we can prove that neither is any other straight line except *FC*; therefore *FC* is perpendicular to *DE*.

Therefore etc. Q. E. D.